To get back to the actual question at hand...

牛乳 is cow`s milk. This has been answered pretty clearly. It`s the kind of milk people normally buy in cartons to drink. Not much of anything to add here.

ミルク is formula or powdered milk. Like the kind you add hot water to for babies.
It is also occasionally used to talk about a flavor - along the lines of "vanilla".
You will not encounter ミルク being used to talk about the type of normal types of milk people drink.

お乳 is not one you hear a lot. Breast milk is 母乳 or just おっぱい. 母乳 is not technical or scientific sounding. It is the regular term used for breast milk or breast feeding. (母乳 vs ミルク) おっぱい is the more childish way of referring to both the breasts and the breast milk. おっぱいを飲む is the standard term for babies at the breast. It`s sort of baby-talk.

For goat milk, 山羊乳 is pretty standard for the liquid form, and 山羊ミルク for powdered.

To add to Nyororin's excellent explanation, ミルク is very often used to refer to what you add to coffee. Whether it's real milk (or cream) or an artificial cream-like liquid product, it's called ミルク at most coffee shops and restaurants.

牛乳, when it's consumed at home, is almost always called as such. However, when it's served at eateries, it's very often called ミルク.

See the drink menu at McDonald's.
レギュラーメニュー | メニュー情報 | McDonald's Japan

お乳, as Nyororin stated, has been getting more obsolete every decade since I was little. I seriously recommend that Japanese learners not use it to refer to breast milk because the word (without the お) is mostly used as a vulgar word for "boobs".

皆さん、本当にどうもありがとうございました。:)

TBH, I don't really understand why anyone would want to teach you お乳 at your level. If I were a teacher, I would only teach it to very advanced students.

Not that it's saying much, but I'd never seen it before the OP. I mean, I could read it because I knew the kanji (although at first I was reading it おにゅう by mistake), but I had never seen that specific word before.

Is it an older word? I'm just guessing so because words like てめぇ are now vulgar, but used to be perfectly normal. The parallel I'm getting at here is お乳 could have been a regular word, but now ちち is sort of a vulgar word like "titties" in English.

It's an old word but then ours is an old country and most words are very old. It was a perfectly normal word when I was a kid in the 60s but I rarely hear it used now. When I do hear it now, it is mostly used by people older than at least 70. I'm sure even today's teenagers know what it means but it would certainly not be in their active vocabulary.

Hi masaegu san, am I right to say that when referring to breast milk, it will be safer to use 母乳 instead of お乳, just in case some people might find it offensive? :)

Yeah, you are. You will get slight stares for saying お乳 in public nowadays.

Thank you, masaegu san. By the way, I have questions regarding 警官、警察 and 警察官. I would like to know which is which when referring to police (a single policeman), police station and policemen (a bunch of them)?

The last part of your question worries me. Most Japanese nouns don't have plural forms and you need to get used to it.

警官　＝　警察官　＝ a policeman/policemen.

警察官 is a tad bit more formal than 警官 but the difference is minimal.

The informal word meaning a policeman is お巡（まわ）りさん.

警察 means the abstract concept "the police" and at the same time is used to mean the "police station" in informal speech.